The solution of problems for multidimensional nonlinear
control systems encounters serious difficulties, which are
both mathematical and technical in nature.
Therefore it is imperative to
develop methods of reduction of nonlinear systems to a
simpler form, for example, decomposition into systems of
lesser dimension. Approaches to reduction are diverse, in
particular, techniques based on approximation methods. We elaborate the most natural and obvious
(in our opinion) approach, which is essentially inherent in
any theory of mathematical entities, for instance, in the
theory of linear spaces, theory of groups, etc.
Reduction in our interpretation is based on
assigning to the initial object an isomorphic object, a
quotient object, and a subobject. In the theory of linear
spaces, for instance, reduction consists in reducing to an
isomorphic linear space, quotient space, and subspace. Strictly speaking, the
exposition of any mathematical theory essentially begins with the
introduction of these reduced objects and determination of
their basic properties in relation to the initial object.
Therefore, we can say that the theory of reduction of
nonlinear control systems discussed here
outlines the elements of the general theory of
such systems. This theory
is purely differential geometric and theoretical group by nature.

A formal definition of these reduced objects that is
suitable for any mathematical theory can be given within
the framework of the theory of categories or the Bourbaki
theory of structures. At the descriptive level every
category (for example, the category of linear spaces or
category of groups) is a class of objects, in which every
object S is a set M with the same sort of predefined
structure. This structure can be interpreted as a collection of a
particular type of relationships between the elements of
the set M. Along with objects, a category also contains
morphisms that implement the object-object
interrelations. If two objects S1 and S2 are defined
on the sets M1 and M2, respectively, then a morphism
f of the object S1 into the object S2 is a
mapping f from M1 to M2 that preserves the
structure of a given kind (i.e., preserves the
corresponding relationships between the elements of the
sets). For example, in the category of linear spaces,
morphisms are linear mappings, while in the category of
groups they are homomorphisms.

For nonlinear control systems dy/dt=f(y,u), it is possible to
construct a category along the following lines. Objects in
this category, which is denoted by NS, are control systems.
Morphisms are defined as follows. Along with some system
S1 described by relations dy/dt=f(y,u), let us consider a control
system S2 described by the relations dx/dt=g(x,v).
A morphism of the system S1 to the system S2 is called
a mapping f of the phase space M of the system
S1 into the phase space L of the system S2 that
carries the solutions (phase trajectories) of the system
S1 into the solutions of the system S2.
Isomorphism in any category is a morphism f that is a one-one
mapping, moreover, the inverse mapping is also
a morphism. If for the objects S1 and S2 there
exists an isomorphism of S1 into S2, then the
objects S1 and S2 are said to be isomorphic.
Isomorphic objects have identical properties within the
framework of the category. For example, in the category of linear
spaces, isomorphisms are linear isomorphisms.

It is appropriate to reduce the control system to an
isomorphic or, alternatively, equivalent system if the latter is simpler in form. For example, a
complicated nonlinear system can be equivalent to a
linear system. In this case, nonlinearity is a `causal
trait,' which vanishes in an equivalent system. Vital
properties of control systems like controllability,
stability, and optimality of solutions are preserved in
changing over to an equivalent system. Therefore, it is
natural to solve any control problem for a simpler
equivalent system and then `carry over' the results thus
obtained to the initial system through an isomorphism.

The concept of a subobject results from our desire to
construct correctly the restriction of an object S1
defined on a set M to a subset N of M. It is
generally not possible to restrict an object S1 to an
arbitrary set. An object S2 defined on a subset
N is called a subobject if the canonical
injection of N to M is a morphism. For example, in
the category of linear space, subobjects are linear
subspaces.

The need for restricting a control system S1, i.e.,
changing over to a subsystem S2 on a subset
N of the set M, is dictated by practical considerations when
the elements of the set M are forced to obey certain
constraints (initial conditions, boundary conditions,
etc.). In such cases, it is natural to restrict a system
S1 to some subset N for which these
constraints are satisfied. The subsystem S2 on
N defines a part of the solutions of the initial system
S1 that lie within N and, in particular, satisfy the
given constraints. Therefore, the solution of a control
problem formulated for a system S1 can be reduced to the
solution of an analogous problem for its subsystem
S2 with phase space of diminished dimension.

While simplification in restriction is achieved via passage to a subset
N of the set M, in factorization it results from
`contraction' of the set M, i.e., passage to the quotient set
M/R over some equivalence relations R. In this passage,
points belonging to one equivalence class are `glued' to
the same point of the quotient set M/R. An object
S2 defined on the quotient set M/R is called the
factor object of the object S1 on the set M if the
canonical projection of Mto M/R is a
morphism. For example, in the category of linear spaces,
factor objects are quotient spaces.

The significance of factorization for the reduction of
control systems lies primarily in the decomposition of
the initial system it generates. More exactly, if a system
S1 has a factor system dz/dt=g(z,v) on some quotient set L=M/R,
then system S1 is equivalent to the system dz/dt=g(z,v), dx/dt=h(z,x,v).
This system by its structure suggests that its solution
z(t), x(t) can be determined as follows. First we
solve of the factor system dz/dt=g(z,v) (for some control v(t))
and then, substituting z(t) into dx/dt=h(z,x,v) , we obtain x(t).
This is the underlying principle of decomposition of
algorithms designed for solving control problems. Let us
note that many properties of a control system
(observability, autonomity, etc.) are determined by its
factor systems of a special type.

The concepts of an isomorphic object, a factor object, and
a subobject may be applied for reducing an object jointly
and in any order. A concrete sequence of transitions to a reduced
system is called the reduction scheme and the number of
transitions is called the reduction depth. To solve a control
problem, a reduction scheme and its depth are chosen from the
conditions of the problem.

A pivotal topic in reduction is the construction of a
mathematical apparatus for determining reduced objects. The
mathematical apparatus includes those concepts that are
invariant to morphisms. An effective instrument for studying reduction in the
category NS and in its different subcategories
is presented by the associative differetial geometric and theoretical group objects:
transformation group, Lie algebra of vector fields, affine distribution, codistribution,
Pfaffian system etc. Objects of such a kind
can be linked with every control system; moreover,
their structure is preserved under the action
of morphisms of the category of control systems. Note for example the following results. The
subsets, on which subsystems may exist, must be invariant manifolds of the associated group. Consequently,
this group must be intransitive so that (nontrivial)
subsystems may exist. On the other hand the associative group must be
imprimitive so that (nontrivial) factor systems may exist.

This approach to the problem of reduction of control systems is presented in the following publications.